The Lagrange top and the fifth Painlev\'e equation

TL;DR

This paper establishes a connection between the Lagrange top, a classical mechanical system, and the fifth Painlevé equation, revealing new insights into their mathematical relationship and parameter interpretations.

Contribution

It demonstrates the equivalence of the Lagrange top with time-dependent inertia to the fifth Painlevé equation, providing a novel interpretation of its parameters as action variables.

Findings

Lagrange top with linearly time-dependent inertia is equivalent to the fifth Painlevé equation.

The harmonic Lagrange top with a quadratic potential is also equivalent to the fifth Painlevé equation under a time-dependent potential.

Two parameters of the fifth Painlevé equation are interpreted as global action variables.

Abstract

We show that the Lagrange top with a linearly time-dependent moment of inertia is equivalent to the degenerate fifth Painlev\'e equation. More generally we show that the harmonic Lagrange top (the ordinary Lagrange top with a quadratic term added in the potential) is equivalent to the fifth Painlev\'e equation when the potential is made time-dependent in an appropriate way. Through this identification two of the parameters of the fifth Painlev\'e equation acquire the interpretation of global action variables. We discuss the relation to the confluent Heun equation, which is the Schr\"odinger equation of the Lagrange top, and discuss the dynamics of $P_V$ from the point of view of the Lagrange top.